Fingerprints of magnetoinduced charge density waves in monolayer graphene beyond half filling

A charge density wave is a condensate of fermions, whose charge density shows a long-range periodic modulation. Such charge density wave can be principally described as a macroscopic quantum state and is known to occur by various formation mechanisms. These are the lattice deforming Peierls transition, the directional, fermionic wave vector orientation prone Fermi surface nesting or the generic charge ordering, which in contrast is associated solely with the undirected effective Coulomb interaction between fermions. In two-dimensional Dirac/Weyl-like systems, the existence of charge density waves is only theoretically predicted within the ultralow energy regime at half filling. Taking graphene as host of two-dimensional fermions described by a Dirac/Weyl Hamiltonian, we tuned indirectly the effective mutual Coulomb interaction between fermions through adsorption of tetracyanoquinodimethane on top in the low coverage limit. We thereby achieved the development of a novel, low-dimensional dissipative charge density wave of Weyl-like fermions, even beyond half filling with additional magneto-induced localization and quantization. This charge density wave appears both, in the electron and the hole spectrum.

I. Supplementary is independent of and solely depends on the screening properties of the system [S1].

II. Supplementary Note 2
All experimentally acquired Raman spectra were analyzed using individual fits with the Lorentz function to the relevant peaks. In order to calculate the TCNQ induced charge carrier density the G-band position was analyzed with respect to the theoretically expected G-band position for pristine graphene. This is necessary as already before TCNQ deposition a non-negligible doping is observable in the pristine samples due to a slightly different coupling to the substrate [S1]. The difference between the position before and after TCNQ deposition is then related to the molecule and can be used to estimate the Fermi energy shift and subsequently the induced charge carrier density using a conversion factor obtained from [S2].

III.
Supplementary Note 3 The discussed CDW signature of a maximal longitudinal resistivity appearing together with unconventional plateaus in transverse conductivity can in principal also be attributed to concurring phenomena such as the quantum Hall insulator (QHI) [S3,S4], the magnetic catalysis [S5-S8] or Zeeman splitting [S9-S11]. However, the unambiguous distinction between these phenomena and a CDW phase is straightforward in the specific case of the = 0 plateau. Regarding the QHI it suffices to consider the extent of the plateau. The QHI only exists and collapses within the lowest Landau level and is followed by the conventional sequence of quantum Hall plateaus (that is = 2 , 6 ... in graphene) with the respective minima in [S3,S4]. Similarly, the magnetic catalysis exclusively reveals plateaus in showing dissipationless longitudinal transport (except at the charge neutrality point where = 0 ), that is, common quantum Hall states.
Furthermore, the mechanism excludes, besides 1 , all other odd plateau values [S6-S9]. Finally, the Zeeman splitting can be tested through the energy width of the 0 plateau at a given magnetic field strength. The energetic splitting magnitude of Zeeman splitting is given by [S12] * ⋅ ⋅ where is the Bohr magneton, is the magnetic field and * is the Landé factor. Under the assumption that * in graphene is the same as in vacuum ( * = 2) [S12] , the magnetic field strength necessary to achieve plateaus with energy widths of 20 meV or 80 meV can be calculated to be 170 T or 700 T, respectively.

IV.
Supplementary Note 4 Measurements of the Hall conductivity of S3 with respect to the charge carrier density for magnetic fields ranging from -12 T to +12 T. The 0 plateau at the charge neutrality point is well pronounced for magnetic field strengths greater than 4 T.
We performed magnetotransport measurements at room temperature and a magnetic field strength of 12 T for the reference sample at 200K (without TCNQ) (cf. Fig. 1a). The sample clearly shows plateaus at 2 in the Hall conductivity , accompanied by minima in the longitudinal resistivity . This presence of quantum Hall states [S13-S16] at moderate temperature demonstrates that our fabrication process is non-invasive and results in high quality graphene samples.
In Fig. 1b we show a summary of the Hall conductivities obtained from magnetotransport measurements performed on Sample S3 with magnetic fields strengths ranging from -12 T to 12 T. The zero conductivity plateau at the CNP as well as the unconventional plateaus at